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Last updated on July 5th, 2025

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Cross Multiply

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We use different methods to solve linear equations in math. One such method is cross multiplication, which is used to compare fractions or solve linear equations involving two variables. In this article, we will learn how to apply this method effectively.

Cross Multiply for Filipino Students
Professor Greenline from BrightChamps

What is Cross-Multiplication?

The common method of multiplying numbers in the form of a fraction is known as cross multiplication. Cross multiplication involves multiplying the numerators of one fraction by the denominator of the other across an equation. This method can be easily applied when solving linear equations with two variables. For example:

 


(a)1x + (b)1y + (c)1 = 0
(a)2x + (b)2y + (c)2 = 0

 


If cross-multiplication is performed correctly, we can quickly obtain the values of x and y by simplifying the equation.
 

Professor Greenline from BrightChamps

How to Cross Multiply Fractions?

Cross multiplication is primarily used with fractions, either to determine the greater fraction or to check if they are equal. This method can help solve equations when dealing with complex fractions. Given below are the different steps involved in this process:

 


For example: 5/8 = 10/16

 

Step 1: To begin, multiply the fraction's numerator on the right by the fraction's denominator on the left. Multiplying 10 × 8 = 80.

 

 

Step 2:  Now, we multiply the fraction's denominator on the right by the fraction's numerator on the left. We can represent 10/16 as 10/1 (since they are equivalent fractions).

 


 
Step 3: After the cross multiplication, always compare the LHS and RHS.
If they are equal, we can conclude that the fractions are equivalent. Here, in this 80 = 80, the fractions are equivalent.
 

Professor Greenline from BrightChamps

What is the Formula for Cross-Multiplication?

The cross-multiplication formula is used for solving linear equations with two variables, as given below: 

 

 

(a)1x + (b)1y + (c)1 = 0
(a)2x + (b)2y + (c)2 = 0

 


The cross-multiplication formula we often apply is:

 


x(b1c1 - b2c2)  = y(c1a2 - c2a1)  = 1(b2a1 - b1a2) 

 

 

How to Derive Cross-Multiplication Formula?


In linear equations with two variables, we derive the cross-multiplication formula by eliminating one variable, often by making the coefficients of that variable equal.
For example:
Let’s consider two linear equations as
(a)1x + (b)1y + (c)1 = 0…(1)
(a)2x + (b)2y + (c)2 = 0…(2)
We solve these equations by making the coefficients of y equal in both equations:
We first multiply equation (1) by b2 and equation (2) by b1:
(b2)(a)1x + (b2)(b)1y + (b2)(c)1 = 0…(3)
(b1)(a)2x + (b1)(b)2y + (b1)(c)2 = 0…(4)
Then, subtract equation (4) from equation (3), 
(b2a1 – b1a2)x + (b2c1 – b1c2) = 0
Isolating x,
x = (b1c2 – b2c1) / (b2a1 – b1a2)…(a)
where (b2a1 – b1a2) ≠ 0
Similarly, solve (1) and (2) for y:
y = (c1a2– c2a1) / (b2a1 – b1a2)…(b)
where (b2a1 – b1a2) ≠ 0
We now combine (a) and (b),
x(b1c1 - b2c2)  = y(c1a2 - c2a1)  = 1(b2a1 - b1a2) 
Thus, we derived the required cross-multiplication formula.
 

Professor Greenline from BrightChamps

How to Solve Linear Equations by Cross Multiplication Method?

Cross-multiplication is especially helpful in linear equations with two variables and is a quick way to find the solution. Solving linear equations with two variables, we apply the cross-multiplication method as given below:
Use the cross-multiplication formula:
x/(b1c2 – b2c1) = y/(c1a2– c2a1) = 1/(b2a1 –  b1a2)
Solve the linear equations:
3x + 4y = 8
2x + y = 5
Now, we convert the equations into standard form:
Since the general form of a linear equation is:
ax + by + c = 0
We rewrite the given equation as:
3x + 4y –8 = 0
2x + y – 5 = 0
When comparing the general form a1x + b1y = – c1 and a2x + b2y = –c2:
a1=3, b1​=4, c1​= −8
a2 = 2, b2=1, c2= −5
Now, apply the cross-multiplication formula:
x/(b1c2 – b2c1) = y/(c1a2– c2a1) = 1/(b2a1 –  b1a2)
Substituting the values:
x/(4) (– 5) – (1) (– 8) = y/(– 8) (2) – (– 5) (3) = 1/ (1) (3) – (4) (2)
Simplifying the fractions:
x/(–20 + 8) = y/(– 16 + 15) = 1/ (3 – 8)
x/(–12) = y/(– 1) = 1/ (– 5)
Isolating x:
Comparing the first and third fractions:
x/ –12 = 1/–5
x = (–12 × 1)/ –5 = 12/5
Isolating y:
Compare the second and third fractions:
y/ – 1 =1/ –5
y = –1 × 1/ –5 = 1/5
x = 12/ 5, y = 1/5
 

Professor Greenline from BrightChamps

Real-Life Applications of Cross-Multiplication

Cross-multiplication in equations helps students determine the value of unknown numbers easily. This technique can be applied to various real-life situations. Here are a few examples of its applications:

 

 

  • Cross-multiplication is used to compare the prices of different items for choosing the cheaper option.

 

  • This method helps adjust the quantities of ingredients required for a recipe without changing the proportions.

 

  • Cross-multiplication is utilized in calculating the time required for a journey at a constant speed.

 

  • We apply cross-multiplication to convert currencies accurately on the basis of their exchange rates.
     
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Common Mistakes and How to Avoid Them in Cross Multiplication

Solving linear equations using cross multiplication is a quick way to obtain results. However, students often make mistakes when solving equations using this method. Here are a few common mistakes and tips to avoid them:

Mistake 1

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Expressing the Equation Incorrectly
 

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Students typically make mistakes in writing the equation properly.
Always write the formula in its standard form for easy reference, i.e., ax + by + c = 0. For example: If given 4x + 5y = 10, always rewrite it as 4x + 5y – 10 = 0.
 

Mistake 2

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 Incorrect Substitution of Values
 

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Some students might swap the given values when applying them to the formula.
When writing the negative value, always use parentheses; write the formulas clearly and check if the values are substituted correctly.
 

Mistake 3

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Not Identifying Coefficients
 

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Students may mistakenly write the values as a1, b1, c1, a2, b2, c2.
Ensure that you write the equations in the standard form and then identify the coefficients.
 

Mistake 4

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 Solving for x and y Incorrectly
 

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In some cases, students might equate the fractions incorrectly when simplifying them.
 While applying the formula for cross multiplication:
x/A = y/ B = 1/ C
Always solve for x and y separately:
x = A ×1/C = A/C
y = B  × 1/C = B/C.
 

Mistake 5

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Not Verifying the Solution

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Students might overlook the verification step to confirm whether the obtained values satisfy the original equations.
 To verify the solution, check by substituting the values of x and y back into the original equations.
 

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Solved Examples of Cross Multiplication

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Problem 1

Solve the following linear equations using the cross-multiplication method: 6x – 3y = 9 8x + 6y = 11

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x = 29/20, y = –1/10
 

Explanation

Rewrite the equations in the form ax + by + c = 0 (standard form):
6x - 3y - 9 = 0
8x + 6y -l 11 = 0
Now, we compare a1x+ b1y+ c1 = 0 and a2x + b2y + c2 = 0:
a1 =6, b1= –3, c1= -9
a2 = 8, b2 = 6, c2= -11
Apply the cross-multiplication formula: 
x/(b1c2 – b2c1) = y/(c1a2– c2a1) = 1/(b2a1 –  b1a2)
Substituting the values:
x/(–3 × –11– 6 × –9) = y/(– 9 × 8 – (– 11 × 6) = 1/ (6 × 6 – 8 × – 3 )
Simplify each term:
(–3 × –11) – (6 × –9) = 33 + 54 = 87
(–9 × 8) – (– 11 × 6) = –72 + 66 = – 6
(6 × 6) – (8 × –3) = 36 + 24 = 60
Thus, the equation simplifies to:
x/87 = y/ –6 = 1/60
Solve for x and y:
x = 87/60 = 29/20
y = –6/60 = –1/10
So, x = 29/20, y = –1/10
 

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Problem 2

Solve the equation: 2/5 = x/ 10

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x = 4
 

Explanation

Apply the cross-multiplication method:
2 × 10 = 5 × x
20 = 5x
Solve for x by dividing both sides by 5:
x = 20/5
x = 4
 

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Max, the Girl Character from BrightChamps

Problem 3

Check if the fractions 3/12 and 5/20 are proportional.

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3/12 and 5/20 are proportional (since both sides are equal.)
 

Explanation

We use cross-multiplication to check if the given fractions are proportional:
Set up the proportion as:
3/12 = 5/20
Now apply cross-multiplication,
3 × 20 = 12 × 5
60 = 60
Since both sides are equal, we can confirm that 3/12 and 5/20 are proportional.
 

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Max, the Girl Character from BrightChamps

Problem 4

Anna needs 6 cups of flour to make 12 pancakes. How many cups of flour are needed to make 18 pancakes?

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9 cups of flour.
 

Explanation

Let the number of cups of flour required to make 18 pancakes be x
Set up the proportion:
6/12 = x/18
Now, apply cross multiplication:
6 × 18 = 12 × x
108 = 12x
Solve for x:
We divide both sides by 12:
x = 108/12 = 9
So, Anna needs 9 cups of flour to make 18 pancakes.
 

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Max, the Girl Character from BrightChamps

Problem 5

A store sells 9 apples for $7. How much would 20 apples cost?

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The cost of 20 apples is approximately $15.56.
 

Explanation

Let y be the cost of 20 apples, 
We first set up the proportion:
9/7 = 20/ y
Apply cross multiplication:
9 × y = 7 × 20
9y = 140
y = 140/9 =15.556
So, the cost of 20 apples is approximately $15.56.
 

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Ray Thinking Deeply About Math Problems

FAQs on Cross Multiplication

1.What do you mean by cross-multiplication?

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2.How can we perform cross-multiplication?

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3.Cite one real-life example of cross-multiplication.

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4.Does the cross-multiplication apply to all fraction equations?

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5.How can we verify the solution?

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6.How can children in Philippines use numbers in everyday life to understand Cross Multiply ?

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7.What are some fun ways kids in Philippines can practice Cross Multiply with numbers?

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8.What role do numbers and Cross Multiply play in helping children in Philippines develop problem-solving skills?

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9.How can families in Philippines create number-rich environments to improve Cross Multiply skills?

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Max, the Girl Character from BrightChamps

Fun Fact

: She loves to read number jokes and games.

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